(a) Find two power series solutions for and express the

Chapter 6, Problem 29E

(choose chapter or problem)

(a) Find two power series solutions for \(y^{\prime \prime}+x y^{\prime}+y=0\) and express the solutions \(y_{1}(x)\) and \(y_{2}(x)\) in terms of summation notation.

(b) Use a CAS to graph the partial sums \(S_{N}(x)\) for y_{1}(x)\). Use N = 2, 3, 5, 6, 8, 10. Repeat using the partial sums \(S_{N}(x)\) for  \(y_{2}(x)\).

(c) Compare the graphs obtained in part (b) with the curve obtained by using a numerical solver. Use the initial-conditions \(y_{1}(0)=1\), \(y_{1}^{\prime}(0)=0\), and \(y_{2}(0)=0\), \(y_{2}^{\prime}(0)=1\).

(d) Reexamine the solution \(y_{1}(x)\) in part (a). Express this series as an elementary function. Then use (5) of Section 4.2 to find a second solution of the equation. Verify that this second solution is the same as the power series solution \(y_{2}(x)\).

Text Transcription:

y^prime\prime+xy^prime+y=0

y_1(x)

y_2(x)

S_N(x)

y_1(0)=1

y_1^prime(0)=0

y_2(0)=0

y_2^prime(0)=1

y_1(x)

y_2(x)